Ergodic Properties of Smooth Systems on Manifolds

流形上光滑系统的遍历性质

• 批准号: 2247572
• 负责人: Adam Kanigowski
• 金额: $ 21.57万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247572&HistoricalAwards=false
• 关键词: Ergodic; Properties; Smooth; Systems; Manifolds

This project focuses on the chaotic properties of smooth dynamical systems, a very active area of current research in mathematical analysis, with connections and applications across the mathematical sciences. Chaotic behavior in dynamical systems is characterized by sensitive dependence on initial conditions, which occurs when small changes in a system’s present state may produce large fluctuations in its future state. This occurs in many natural systems, such as the human heart, fluid flows, and global weather patterns. The PI plans to develop a general framework and new approaches for understanding chaotic behavior in a large class of dynamical systems. Activities in the project will lead to progress in our understanding of fundamental dynamical phenomena, with possible consequences and applications in other mathematical fields, such as geometry and number theory, and other scientific areas, such as physics and economics. The project will also involve the training of several students and postdocs. This project is part of a program of research studying ergodic and statistical properties of smooth systems on manifolds and their interactions with geometry and number theory, with three main areas of focus. One area concerns some classical properties that are expected to hold for chaotic systems: K and Bernoulli properties, quantitative mixing (and higher order mixing) and limit theorems. The PI will investigate relations between these properties and their appearance for partially hyperbolic (or non-uniformly partially hyperbolic) systems, and continue developing a geometric framework for problems related to quantitative mixing and the Bernoulli property. The PI also plans to construct examples of exotic dynamical systems with new ergodic and statistical behavior. A second research direction relates to recent developments on parabolic systems, of not necessarily algebraic origin. Despite recent progress, many fundamental questions for such systems are still open, for instance, the Rokhlin problem on higher order mixing. The PI will build on techniques from his earlier work and try to develop a general theory for ergodic properties of parabolic systems. A third part of the project will continue the PI’s research on sparse equidistribution problems, using methods from dynamics (such as quantitative equidistribution and mixing) and analytic number theory (such as sieve methods and exponential sums).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的重点是光滑动力系统的混沌特性，这是当前数学分析研究中非常活跃的领域，与数学科学的联系和应用。动力系统中的混沌行为的特征在于对初始条件的敏感依赖，当系统当前状态的微小变化可能会在其未来状态中产生大的波动时，就会发生这种情况。 这种情况发生在许多自然系统中，例如人类心脏、流体流动和全球天气模式。 PI计划开发一个通用框架和新方法来理解一大类动力系统中的混沌行为。 该项目的活动将导致我们对基本动力学现象的理解取得进展，并可能在其他数学领域（如几何和数论）以及其他科学领域（如物理和经济学）产生影响和应用。该项目还将涉及培训若干学生和博士后。 该项目是研究流形上光滑系统的遍历和统计特性及其与几何和数论的相互作用的研究计划的一部分，重点有三个主要领域。一个领域涉及一些经典的性质，预计将举行混沌系统：K和伯努利性质，定量混合（和高阶混合）和极限定理。PI将研究这些属性之间的关系，并为部分双曲（或非均匀部分双曲）系统的外观，并继续发展与定量混合和伯努利性质相关的问题的几何框架。 PI还计划构建具有新遍历和统计行为的奇异动力系统的示例。 第二个研究方向涉及抛物系统的最新发展，不一定是代数起源。 尽管最近的进展，这类系统的许多基本问题仍然是开放的，例如，高阶混合的Rokhlin问题。 PI将建立在他早期工作的技术基础上，并试图为抛物系统的遍历性建立一个通用理论。 该项目的第三部分将继续PI对稀疏均匀分布问题的研究，使用动力学方法（如定量均匀分布和混合）和解析数论（如筛法和指数和）。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。